A population of 50 flies is expected to double every week, leading to a function of the form f(x) = 50(2)^x, where x represents the number of weeks that have passed. When will this population reach 500? Trying to solve this problem leads to 500 = 50(2)^x Dividing both sides by 50 to isolate the exponential leads to 10 = 2 ^x While we have set up exponential models and used them to make predictions, you may have noticed that solving exponential equations has not yet been mentioned. The reason is simple: none of the algebraic tools discussed so far are sufficient to solve exponential equations. Consider the equation 2 ^x = 10above. We know that 2³ = 8 and 2⁴ = 16, so it is clear that x must be some value between 3 and 4 since g(x) = 2^x is increasing. We could use technology to create a table of values or graph to better estimate the solution. From the graph, we could better estimate the solution to be around 3.3. This result is still fairly unsatisfactory, and since the exponential function is one-to-one, it would be great to have an inverse function. None of the functions we have already discussed would serve as an inverse function and so we must introduce a new function, named log as the inverse of an exponential function. Since exponential functions have different bases, we will define corresponding logarithms of different bases as well.

Logarithm

The logarithm (base b) function, written log_b(x), is the inverse of the exponential function (base b), b^x. Since the logarithm and exponential are inverses, it follows that:

Properties of Logs: Inverse Properties

[latex-display]\displaystyle{{log}_{{b}}{({b}^{{x}})}}={x}[/latex-display] [latex-display]\displaystyle{b}^{{{{log}_{{b}}{x}}}}={x}[/latex-display] Recall also from the definition of an inverse function that if f(a) = c, then f^–1(c) = a. Applying this to the exponential and logarithmic functions:

Logarithm Equivalent to an Exponential

The statement b^a = c is equivalent to the statement log_b(c) = a. Alternatively, we could show this by starting with the exponential function c = b^a, then taking the log base b of both sides, giving log_b(c) = log_bb^a. Using the inverse property of logs we see that log_b(c) = a. Since log is a function, it is most correctly written as log _b (c), using parentheses to denote function evaluation, just as we would with f(c). However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written as log_bc.

Example 1

Write these exponential equations as logarithmic equations:

• 2³ = 8
• 5² = 25
• [latex]\displaystyle{10}^{{-{{4}}}}=\frac{{1}}{{10000}}[/latex]

• [latex]\displaystyle{2}^{{3}}={8}[/latex]
• [latex]\displaystyle{5}^{{2}}={25}[/latex]
• [latex]\displaystyle{10}^{{-{4}}}=\frac{{1}}{{10000}}[/latex]

 

Example 2

Write these logarithmic equations as exponential equations:

• [latex]\displaystyle{{log}_{{6}}{(sqrt{{6}})}}=\frac{{1}}{{2}}[/latex]
• [latex]\displaystyle{{log}_{{3}}{({9})}}={2}[/latex]

 

• [latex]\displaystyle{{log}_{{6}}{(sqrt{{6}})}}=\frac{{1}}{{2}}[/latex]
• [latex]\displaystyle{{log}_{{3}}{({9})}}={2}[/latex]

Try it Now 1

Write the exponential 4 ² = 16 equation as a logarithmic equation. By establishing the relationship between exponential and logarithmic functions, we can now solve basic logarithmic and exponential equations by rewriting.

Example 3

Solve log ₄(x) = 2 for x. By rewriting this expression as an exponential, 4 ² = x, so x = 16

Example 4

Solve 2 ^x = 10 for x. By rewriting this expression as a logarithm, we get x = log₂(10) While this does define a solution, and an exact solution at that, you may find it somewhat unsatisfying since it is difficult to compare this expression to the decimal estimate we made earlier. Also, giving an exact expression for a solution is not always useful—often we really need a decimal approximation to the solution. Luckily, this is a task calculators and computers are quite adept at. Unluckily for us, most calculators and computers will only evaluate logarithms of two bases. Happily, this ends up not being a problem, as we'll see briefly.

Common and Natural Logarithms

The common log is the logarithm with base 10, and is typically written log(x). The natural log is the logarithm with base e, and is typically written log(e).

Example 5

Evaluate log(1000) using the definition of the common log.

Values of the Common Log
number	number as exponential	log(number)
1000	10 3	3
100	10 2	2
10	10 1	1
1	10 0	0
0.1	10 –1	–1
0.01	10 –2	–2
0.001	10 –3	–3

To evaluate log(1000), we can say x = log(1000), then rewrite into exponential form using the common log base of 10. 10 ^x = 1000 From this, we might recognize that 1000 is the cube of 10, so x = 3. We also can use the inverse property of logs to write log ₁₀(10³) = 3

Try it Now 2

Evaluate log(1000000).

Example 6

Evaluate [latex]\displaystyle{ln{{(sqrt{{e}})}}}[/latex]. We can rewrite as [latex]\displaystyle{ln{{(sqrt{{e}})}}}[/latex].

Example 7

Evaluate log(500) using your calculator or computer. Using a computer, we can evaluate log(500) ≈ 2.69897 To utilize the common or natural logarithm functions to evaluate expressions like log ₂(10), we need to establish some additional properties.

Properties of Logs: Exponent Property

log _b(A^r) = rlog_b(A) To show why this is true, we offer a proof. Since the logarithmic and exponential functions are inverses, [latex]\displaystyle{b}^{{{{log}_{{b}}{A}}}}={A}[/latex]. So [latex-display]\displaystyle{A}^{{r}}={({b}^{{{{log}_{{b}}{A}}}})}^{{r}}[/latex-display] Utilizing the exponential rule that states [latex]\displaystyle{({x}^{{p}})}^{{q}}={x}^{{{p}{q}}}[/latex], [latex-display]\displaystyle{A}^{{r}}={({b}^{{{{log}_{{b}}{A}}}})}^{{r}}={b}^{{{r}{{log}_{{b}}{A}}}}[/latex-display] So then [latex-display]\displaystyle{{log}_{{b}}{({A}^{{r}})}}={{log}_{{b}}{({b}^{{{r}{{log}_{{b}}{A}}}})}}[/latex-display] Again utilizing the inverse property on the right side yields the result [latex-display]\displaystyle{{log}_{{b}}{({A}^{{r}})}}={r}{{log}_{{b}}{A}}[/latex-display]

Example 8

Rewrite log ₃(25) using the exponent property for logs. Since 25 = 5 ², log₃(25) = log₃(5²) = 2log₃5

Example 9

Rewrite 4ln( x) using the exponent property for logs. Using the property in reverse, 4ln( x) = ln(x⁴)

Try it Now 3

Rewrite using the exponent property for logs: [latex]\displaystyle{ln{{(\frac{{1}}{{{x}^{{2}}}})}}}[/latex]. The exponent property allows us to find a method for changing the base of a logarithmic expression.

Properties of Logs: Change of Base

[latex-display]\displaystyle{{log}_{{b}}{({A})}}=\frac{{{{log}_{{c}}{({A})}}}}{{{{log}_{{c}}{({b})}}}}[/latex-display]

Proof

Let log _b(A) = x. Rewriting as an exponential gives b^x = A. Taking the log base c of both sides of this equation gives log_cb^x = log_cA. Now utilizing the exponent property for logs on the left side, xlog_cb = log_cA Dividing, we obtain [latex-display]\displaystyle{x}=\frac{{{{log}_{{c}}{A}}}}{{{{log}_{{c}}{b}}}}[/latex-display] With this change of base formula, we can finally find a good decimal approximation to our question from the beginning of the section.

Example 10

Evaluate log ₂(10) using the change of base formula. According to the change of base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base e: [latex-display]\displaystyle{{log}_{{{2}}}{10}}=\frac{{{{log}_{{e}}{10}}}}{{{{log}_{{e}}{2}}}}=\frac{{{ln{{10}}}}}{{{ln{{2}}}}}[/latex-display] Using our calculators to evaluate this, [latex-display]\displaystyle\frac{{{ln{{10}}}}}{{{ln{{2}}}}}approxfrac{{2.30259}}{{0.69315}}approx{3.3219}[/latex-display] This finally allows us to answer our original question—the population of flies we discussed at the beginning of the section will take 3.32 weeks to grow to 500.

Example 11

Evaluate log ₅(100) using the change of base formula. We can rewrite this expression using any other base. If our calculators are able to evaluate the common logarithm, we could rewrite using the common log, base 10. [latex-display]\displaystyle{{log}_{{5}}{({100})}}=\frac{{{{log}_{{{10}}}{100}}}}{{{{log}_{{{10}}}{5}}}}approxfrac{{2}}{{0.69897}}={2.861}[/latex-display] While we were able to solve the basic exponential equation 2 ^x = 10 by rewriting in logarithmic form and then using the change of base formula to evaluate the logarithm, the proof of the change of base formula illuminates an alternative approach to solving exponential equations.

Important Topics of this Section

• The Logarithmic function as the inverse of the exponential function
• Writing logarithmic & exponential expressions
• Properties of logs
  • Inverse properties
  • Exponential properties
  • Change of base
• Common log
• Natural log

Try it Now Answers

1. log₄(16) = 2 = log₄4² = 2log₄4
2. 6
3. –2ln(x)

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David Lippman and Melonie Rasmussen, Open Text Bookstore, Precalculus: An Investigation of Functions, "Chapter 4: Exponential and Logarithmic Functions," licensed under a CC BY-SA 3.0 license.